= Gamma Distribution =

The '''gamma distribution''' is a non-negative continuous probability distribution function.

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== Description ==

The distribution is characterized by '''shape''' and '''scale''' parameters, both of which are strictly positive. A random variable distributed by this function is expressed as ''X ~ Gamma(α, β)''.

The distribution is given as:

{{attachment:gamma.svg}}

for any ''x ∈ [0,∞)'', and where ''Γ'' is the [[Analysis/GammaFunction|gamma function]].

In some cases, the distribution is parameterized to a '''rate''' (''θ'') instead of scale (''β''). Note that rate is the inverse of scale, i.e. ''θ = 1/β'', so the substitution is very simple to apply. The distribution becomes:

{{attachment:gammarate.svg}}

The [[Analysis/ExponentialDistribution|exponential distribution]] is equivalent to a gamma distribution parameterized for rate with shape fixed at 1:  ''Gamma(1, θ) = Exp(θ)''.

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== Moments ==

The [[Analysis/ExpectedValue|expected value]] is given as ''E[X] = αβ = α/θ''.

[[Analysis/Variance|Variance]] is given as ''Var[X] = αβ^2^ = α/(θ^2^)''.



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